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CPS1930 M. Kayanan et al.

If for some such that ̂(−1) ≠ 0, then is the matrix formed by
removing the column from −1. Note that this is the modification done in
LARS algorithm to obtain LASSO estimates. Then the residual related to the
current step is calculated as

and then, move to the next step where +1 is the value of such that
.
Repeat this step until = 1.
2.2. LARS-EN algorithm
The LARS-EN algorithm (Zou & Hastie (2003)) is used to obtain Elastic net
estimates, and it is also a modified version of the LARS algorithm. In LARS-EN
algorithm, the equiangular vector of the LARS algorithm in the equation (8)
is replaced by incorporating RE as follows:
= ( ( + )) ′−1, (10)
−1
′
′
′
and the rest of the steps are similar to the algorithm status above.
2.3. LARS-LEnet algorithm
According to Liu (1993), the LE is defined as
−1
−1
̂ = ( + ) ( + )(′) (11)
′
′
′
where 0 < < 1 is the shrinkage parameter.
Now we propose LARS-LEnet algorithm by incorporating LE in the
LARS algorithm. Here we modify the equiangular vector of the LARS
algorithm in the equation (8) as:
,
(12) and all other steps described in section 2.1 are the same.
2.4. Performance evaluation
The performance of LEnet, Enet and LASSO estimators were compared
using Root Mean Square Error (RMSE) sense, which is the expected prediction
error. The RMSE is defined as
= ( − ̂)( − ̂) (13)
′
where (,) denotes new data that are not used to obtain the coefficient
estimates ̂.
In this study, we considered two real-world examples, namely the Prostate
Cancer Data (Stamey et al. (1989)), and the UScrime dataset (Venables &
Ripley (1999)), to compare the performance of the three estimators LEnet, Enet
and LASSO.
In the Prostate Cancer Data, the predictors are eight clinical measures: log
cancer volume (lcavol), log prostate weight (lweight), age, log of the amount
of benign prostatic hyperplasia (lbph), seminal vesicle invasion (svi), log
capsular penetration (lcp), Gleason score (gleason) and percentage Gleason

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